query rewriting in dl-litehorn(hn)ebotoeva/presentations/boac-dl...motivationthe dl dl-lite(hn) horn...

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Motivation The DL DL-Lite (HN) horn Knowledge Base Satisfiability Query Answering Conclusions Query Rewriting in DL-Lite (HN ) horn Elena Botoeva, Alessandro Artale, and Diego Calvanese KRDB Research Centre Free University of Bozen-Bolzano I-39100 Bolzano, Italy Description Logics Workshop, 2010 Botoeva, Artale, Calvanese Query Rewriting in DL-Lite (HN) horn 1/25

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Page 1: Query Rewriting in DL-Litehorn(HN)ebotoeva/presentations/boac-DL...MotivationThe DL DL-Lite(HN) horn Knowledge Base Satis abilityQuery AnsweringConclusions Query Answering by Rewriting

Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Rewriting in DL-Lite(HN )horn

Elena Botoeva, Alessandro Artale, and Diego Calvanese

KRDB Research CentreFree University of Bozen-Bolzano

I-39100 Bolzano, Italy

Description Logics Workshop, 2010

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

1/25

Page 2: Query Rewriting in DL-Litehorn(HN)ebotoeva/presentations/boac-DL...MotivationThe DL DL-Lite(HN) horn Knowledge Base Satis abilityQuery AnsweringConclusions Query Answering by Rewriting

Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation

2 The DL DL-Lite(HN)horn

3 Knowledge Base Satisfiability

4 Query Answering

5 Conclusions

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

2/25

Page 3: Query Rewriting in DL-Litehorn(HN)ebotoeva/presentations/boac-DL...MotivationThe DL DL-Lite(HN) horn Knowledge Base Satis abilityQuery AnsweringConclusions Query Answering by Rewriting

Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Motivation: Ontology-Based Data Access

• Ontologies are used for accessing data

query

Application

response

Application 1 Application 2

query

response

Data source 1

Data source 2

Data source 3

• An ontology provides a high-level conceptual view of informationsources

• Data sources can be queried through ontologies

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

3/25

Page 4: Query Rewriting in DL-Litehorn(HN)ebotoeva/presentations/boac-DL...MotivationThe DL DL-Lite(HN) horn Knowledge Base Satis abilityQuery AnsweringConclusions Query Answering by Rewriting

Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Motivation: Ontology-Based Data Access

• Ontologies are used for accessing data

Data source 1

Data source 2

Data source 3

C1

A2

R1

C2

C3

A1

query

Ontology

Application

response

Application

C1

A2

R1

C2

C3

A1

Ontology

• An ontology provides a high-level conceptual view of informationsources

• Data sources can be queried through ontologies

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

3/25

Page 5: Query Rewriting in DL-Litehorn(HN)ebotoeva/presentations/boac-DL...MotivationThe DL DL-Lite(HN) horn Knowledge Base Satis abilityQuery AnsweringConclusions Query Answering by Rewriting

Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering by Rewriting

• We want to compute certain answers to a query

• Rewriting approach:

1 Rewrite the query using the constraints in the ontology2 Evaluate the rewritten query over the database

Reasoning

Rewritten Query

Query Result

Reasoning

DataSource

LogicalSchema

Schema /Ontology

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

4/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering by Rewriting

• We want to compute certain answers to a query

• Rewriting approach:

1 Rewrite the query using the constraints in the ontology2 Evaluate the rewritten query over the database

Reasoning

Rewritten Query

Query Result

Reasoning

DataSource

LogicalSchema

Schema /Ontology

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

4/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering by Rewriting: Example

Ontology: O = {PhDStudent v Student}Database: DBA = {PhDStudent(john)}Query: q(x)← Student(x)

• The rewriting of q:qucq(x)← Student(x)qucq(x)← PhDStudent(x)

• By evaluating the rewriting over the ABox viewed as a DB :eval(qucq, DBA) = {john} = ans(q, 〈O,DBA〉)

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

5/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering by Rewriting: Example

Ontology: O = {PhDStudent v Student}Database: DBA = {PhDStudent(john)}Query: q(x)← Student(x)

• The rewriting of q:qucq(x)← Student(x)qucq(x)← PhDStudent(x)

• By evaluating the rewriting over the ABox viewed as a DB :eval(qucq, DBA) = {john} = ans(q, 〈O,DBA〉)

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

5/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

FOL Rewritable Logics

• Such a rewriting approach can be applied only to FOL rewritablelogics.

• DL-Lite is a family of logics that has been shown to enjoy FOLrewritability:

I DL-LiteR, DL-LiteF , DL-LiteA

• Extended DL-Lite family: additional constructs have been proposed

I DL-Lite(HN )horn is the most interesting logic

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

6/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

FOL Rewritable Logics

• Such a rewriting approach can be applied only to FOL rewritablelogics.

• DL-Lite is a family of logics that has been shown to enjoy FOLrewritability:

I DL-LiteR, DL-LiteF , DL-LiteA

• Extended DL-Lite family: additional constructs have been proposed

I DL-Lite(HN )horn is the most interesting logic

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

6/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

FOL Rewritable Logics

• Such a rewriting approach can be applied only to FOL rewritablelogics.

• DL-Lite is a family of logics that has been shown to enjoy FOLrewritability:

I DL-LiteR, DL-LiteF , DL-LiteA

• Extended DL-Lite family: additional constructs have been proposed

I DL-Lite(HN )horn is the most interesting logic

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

6/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation

2 The DL DL-Lite(HN)horn

3 Knowledge Base Satisfiability

4 Query Answering

5 Conclusions

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

7/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN )horn

In this work we consider the logic DL-Lite(HN )horn :

• The most expressive tractable variant of DL-Lite [ACKZ09].

• Extends DL-Lite withI role inclusions H

• hasConfPaper v hasPublication

I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥

I horn inclusions horn• Student u ≥1 teaches v PhDStudent

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

8/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN )horn

In this work we consider the logic DL-Lite(HN )horn :

• The most expressive tractable variant of DL-Lite [ACKZ09].

• Extends DL-Lite with

I role inclusions H• hasConfPaper v hasPublication

I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥

I horn inclusions horn• Student u ≥1 teaches v PhDStudent

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

8/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN )horn

In this work we consider the logic DL-Lite(HN )horn :

• The most expressive tractable variant of DL-Lite [ACKZ09].

• Extends DL-Lite withI role inclusions H

• hasConfPaper v hasPublication

I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥

I horn inclusions horn• Student u ≥1 teaches v PhDStudent

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

8/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN )horn

In this work we consider the logic DL-Lite(HN )horn :

• The most expressive tractable variant of DL-Lite [ACKZ09].

• Extends DL-Lite withI role inclusions H

• hasConfPaper v hasPublication

I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥

I horn inclusions horn• Student u ≥1 teaches v PhDStudent

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

8/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN )horn

In this work we consider the logic DL-Lite(HN )horn :

• The most expressive tractable variant of DL-Lite [ACKZ09].

• Extends DL-Lite withI role inclusions H

• hasConfPaper v hasPublication

I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥

I horn inclusions horn• Student u ≥1 teaches v PhDStudent

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

8/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Questions Addressed by Our Work

For the logic DL-Lite(HN )horn :

• Can we check ontology satisfiability by relying on RDB technology?

• Can we answer queries by relying on RDB technology?

• Can we extend the practical algorithms developed forthe simpler DL-Lite logics?

• What is the complexity of such algorithms?

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

9/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN )horn : Syntax

Concept and role expressions

B ::= ⊥ | A | ≥k R

R ::= P | P−

TBox assertions

B1 u · · · u Bn v BR1 v R2

Dis(R1,R2)

Restriction to ensure FOL rewritability:

if R has a proper sub-role, then ≥k R with k ≥ 2does not occur in the lhs of concept inclusions.

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

10/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN )horn : Syntax

Concept and role expressions

B ::= ⊥ | A | ≥k R

R ::= P | P−

TBox assertions

B1 u · · · u Bn v BR1 v R2

Dis(R1,R2)

Restriction to ensure FOL rewritability:

if R has a proper sub-role, then ≥k R with k ≥ 2does not occur in the lhs of concept inclusions.

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

10/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN )horn TBox

• basic concept inclusion

≥1 hasPublication− v Publication

• role inclusion

hasConfPaper v hasPublication

• number restrictions

PhDStudent v ≥2 hasConfPaper

• horn inclusion

Student u ≥1 teaches v PhDStudent

• local functionality assertion

PhDStudent u ≥2 teaches v ⊥

• global functionality assertion

≥2 teaches− v ⊥

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

11/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN )horn TBox

• basic concept inclusion

≥1 hasPublication− v Publication

• role inclusion

hasConfPaper v hasPublication

• number restrictions

PhDStudent v ≥2 hasConfPaper

• horn inclusion

Student u ≥1 teaches v PhDStudent

• local functionality assertion

PhDStudent u ≥2 teaches v ⊥

• global functionality assertion

≥2 teaches− v ⊥

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

11/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN )horn TBox

• basic concept inclusion

≥1 hasPublication− v Publication

• role inclusion

hasConfPaper v hasPublication

• number restrictions

PhDStudent v ≥2 hasConfPaper

• horn inclusion

Student u ≥1 teaches v PhDStudent

• local functionality assertion

PhDStudent u ≥2 teaches v ⊥

• global functionality assertion

≥2 teaches− v ⊥

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

11/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN )horn TBox

• basic concept inclusion

≥1 hasPublication− v Publication

• role inclusion

hasConfPaper v hasPublication

• number restrictions

PhDStudent v ≥2 hasConfPaper

• horn inclusion

Student u ≥1 teaches v PhDStudent

• local functionality assertion

PhDStudent u ≥2 teaches v ⊥

• global functionality assertion

≥2 teaches− v ⊥

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

11/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN )horn TBox

• basic concept inclusion

≥1 hasPublication− v Publication

• role inclusion

hasConfPaper v hasPublication

• number restrictions

PhDStudent v ≥2 hasConfPaper

• horn inclusion

Student u ≥1 teaches v PhDStudent

• local functionality assertion

PhDStudent u ≥2 teaches v ⊥

• global functionality assertion

≥2 teaches− v ⊥

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

11/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN )horn TBox

• basic concept inclusion

≥1 hasPublication− v Publication

• role inclusion

hasConfPaper v hasPublication

• number restrictions

PhDStudent v ≥2 hasConfPaper

• horn inclusion

Student u ≥1 teaches v PhDStudent

• local functionality assertion

PhDStudent u ≥2 teaches v ⊥

• global functionality assertion

≥2 teaches− v ⊥

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

11/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation

2 The DL DL-Lite(HN)horn

3 Knowledge Base Satisfiability

4 Query Answering

5 Conclusions

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

12/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

• T : Student u Professor v ⊥, PhDStudent v Student

A : PhDStudent(john), Professor(john)

• T : Dis(teaches, attends)

A : teaches(john, cl), attends(john, cl)

• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)

⇒ We need to calculate closure of NIs w.r.t. PIs

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

• T : Student u Professor v ⊥, PhDStudent v Student

A : PhDStudent(john), Professor(john)

• T : Dis(teaches, attends)

A : teaches(john, cl), attends(john, cl)

• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)

⇒ We need to calculate closure of NIs w.r.t. PIs

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

13/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

• T : Student u Professor v ⊥, PhDStudent v Student

A : PhDStudent(john), Professor(john)

• T : Dis(teaches, attends)

A : teaches(john, cl), attends(john, cl)

• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)

⇒ We need to calculate closure of NIs w.r.t. PIs

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

• T : Student u Professor v ⊥, PhDStudent v Student

A : PhDStudent(john), Professor(john)

• T : Dis(teaches, attends)

A : teaches(john, cl), attends(john, cl)

• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)

⇒ We need to calculate closure of NIs w.r.t. PIs

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

13/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

• T : Student u Professor v ⊥, PhDStudent v Student

A : PhDStudent(john), Professor(john)

• T : Dis(teaches, attends)

A : teaches(john, cl), attends(john, cl)

• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)

⇒ We need to calculate closure of NIs w.r.t. PIs

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

13/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability Algorithm

We reduce the problem to FOL query evaluation.

Algorithm for checking KB satisfiability

1 Calculate the closure of NIs.

2 Translate the closure into a UCQ qunsat asking for violation of some NI.

3 Evaluate qunsat over the ABox (viewed as a DB).

I if eval(qunsat ,DBA) = ∅, then the KB is satisfiable;I otherwise the KB is unsatisfiable.

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability Algorithm

We reduce the problem to FOL query evaluation.

Algorithm for checking KB satisfiability

1 Calculate the closure of NIs.

2 Translate the closure into a UCQ qunsat asking for violation of some NI.

3 Evaluate qunsat over the ABox (viewed as a DB).

I if eval(qunsat ,DBA) = ∅, then the KB is satisfiable;I otherwise the KB is unsatisfiable.

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

14/25

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs

• every NI is in cln(T ).

• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent

}⇒

add to cln(T ) : Professor u Student u ≥1 teaches v ⊥

• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches

}⇒

add to cln(T ): PhDStudent u FullProfessor v ⊥

• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends

}⇒

add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs

• every NI is in cln(T ).

• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent

}⇒

add to cln(T ) : Professor u Student u ≥1 teaches v ⊥

• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches

}⇒

add to cln(T ): PhDStudent u FullProfessor v ⊥

• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends

}⇒

add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs

• every NI is in cln(T ).

• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent

}⇒

add to cln(T ) : Professor u Student u ≥1 teaches v ⊥

• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches

}⇒

add to cln(T ): PhDStudent u FullProfessor v ⊥

• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends

}⇒

add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs

• every NI is in cln(T ).

• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent

}⇒

add to cln(T ) : Professor u Student u ≥1 teaches v ⊥

• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches

}⇒

add to cln(T ): PhDStudent u FullProfessor v ⊥

• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends

}⇒

add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs

• every NI is in cln(T ).

• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent

}⇒

add to cln(T ) : Professor u Student u ≥1 teaches v ⊥

• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches

}⇒

add to cln(T ): PhDStudent u FullProfessor v ⊥

• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends

}⇒

add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs

• every NI is in cln(T ).

• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent

}⇒

add to cln(T ) : Professor u Student u ≥1 teaches v ⊥

• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches

}⇒

add to cln(T ): PhDStudent u FullProfessor v ⊥

• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends

}⇒

add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs

• every NI is in cln(T ).

• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent

}⇒

add to cln(T ) : Professor u Student u ≥1 teaches v ⊥

• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches

}⇒

add to cln(T ): PhDStudent u FullProfessor v ⊥

• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends

}⇒

add to cln(T ): Professor u ≥1 registeredTo v ⊥

• · · ·

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs

• every NI is in cln(T ).

• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent

}⇒

add to cln(T ) : Professor u Student u ≥1 teaches v ⊥

• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches

}⇒

add to cln(T ): PhDStudent u FullProfessor v ⊥

• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends

}⇒

add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Translation to FOL Queries

• Professor u Student v ⊥ ⇒∃x .Professor(x) ∧ Student(x).

• ≥2 teaches− v ⊥ ⇒∃x1, x2, y .teaches(x1, y) ∧ teaches(x2, y) ∧ x1 6= x2.

• Dis(attends, teaches) ⇒∃x , y .attends(x , y) ∧ teaches(x , y).

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Translation to FOL Queries

• Professor u Student v ⊥ ⇒∃x .Professor(x) ∧ Student(x).

• ≥2 teaches− v ⊥ ⇒∃x1, x2, y .teaches(x1, y) ∧ teaches(x2, y) ∧ x1 6= x2.

• Dis(attends, teaches) ⇒∃x , y .attends(x , y) ∧ teaches(x , y).

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Translation to FOL Queries

• Professor u Student v ⊥ ⇒∃x .Professor(x) ∧ Student(x).

• ≥2 teaches− v ⊥ ⇒∃x1, x2, y .teaches(x1, y) ∧ teaches(x2, y) ∧ x1 6= x2.

• Dis(attends, teaches) ⇒∃x , y .attends(x , y) ∧ teaches(x , y).

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

KB Satisfiability: Complexity of the Algorithm

• Optimal data complexity: in AC0 (follows from FOL rewritability)

• Combined complexity: exponentialI Worst case: the size of cln(T ) is exponential in the size of the TBoxT = { A′1 v A1, . . . , A′n v An, A1 u · · · u An v ⊥ }.

Notice, that the problem is PTime [ACKZ09].

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

KB Satisfiability: Complexity of the Algorithm

• Optimal data complexity: in AC0 (follows from FOL rewritability)

• Combined complexity: exponentialI Worst case: the size of cln(T ) is exponential in the size of the TBoxT = { A′1 v A1, . . . , A′n v An, A1 u · · · u An v ⊥ }.

Notice, that the problem is PTime [ACKZ09].

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

KB Satisfiability: Complexity of the Algorithm

• Optimal data complexity: in AC0 (follows from FOL rewritability)

• Combined complexity: exponentialI Worst case: the size of cln(T ) is exponential in the size of the TBoxT = { A′1 v A1, . . . , A′n v An, A1 u · · · u An v ⊥ }.

Notice, that the problem is PTime [ACKZ09].

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation

2 The DL DL-Lite(HN)horn

3 Knowledge Base Satisfiability

4 Query Answering

5 Conclusions

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x)← hasPublication(x , y) ∧ Publication(y)

TBox T = {≥1 hasPublication− v Publication

hasConfPaper v hasPublication

PhDStudent v ≥2 hasConfPaper

Student u ≥1 teaches v PhDStudent

}

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication

hasConfPaper v hasPublication

PhDStudent v ≥2 hasConfPaper

Student u ≥1 teaches v PhDStudent

⇓ ≥1 hasPublication− v Publication

q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)

⇓ unify the atoms

q3(x)← hasPublication(x , y)⇓ remove unbound variables

q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication

q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper

q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper

q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent

q7(x)← Student(x) ∧ E1teaches(x)

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication

hasConfPaper v hasPublication

PhDStudent v ≥2 hasConfPaper

Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication

q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)

⇓ unify the atoms

q3(x)← hasPublication(x , y)⇓ remove unbound variables

q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication

q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper

q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper

q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent

q7(x)← Student(x) ∧ E1teaches(x)

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication

hasConfPaper v hasPublication

PhDStudent v ≥2 hasConfPaper

Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication

q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)

⇓ unify the atoms

q3(x)← hasPublication(x , y)

⇓ remove unbound variables

q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication

q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper

q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper

q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent

q7(x)← Student(x) ∧ E1teaches(x)

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication

hasConfPaper v hasPublication

PhDStudent v ≥2 hasConfPaper

Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication

q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)

⇓ unify the atoms

q3(x)← hasPublication(x , y)⇓ remove unbound variables

q3(x)← E1hasPublication(x)

⇓ hasConfPaper v hasPublication

q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper

q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper

q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent

q7(x)← Student(x) ∧ E1teaches(x)

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication

hasConfPaper v hasPublication

PhDStudent v ≥2 hasConfPaper

Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication

q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)

⇓ unify the atoms

q3(x)← hasPublication(x , y)⇓ remove unbound variables

q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication

q4(x)← E1hasConfPaper(x)

⇓ ≥2 hasConfPaper v ≥1 hasConfPaper

q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper

q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent

q7(x)← Student(x) ∧ E1teaches(x)

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication

hasConfPaper v hasPublication

PhDStudent v ≥2 hasConfPaper

Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication

q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)

⇓ unify the atoms

q3(x)← hasPublication(x , y)⇓ remove unbound variables

q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication

q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper

q5(x)← E2hasConfPaper(x)

⇓ PhDStudent v ≥2 hasConfPaper

q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent

q7(x)← Student(x) ∧ E1teaches(x)

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication

hasConfPaper v hasPublication

PhDStudent v ≥2 hasConfPaper

Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication

q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)

⇓ unify the atoms

q3(x)← hasPublication(x , y)⇓ remove unbound variables

q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication

q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper

q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper

q6(x)← PhDStudent(x)

⇓ Student u ≥1 teaches v PhDStudent

q7(x)← Student(x) ∧ E1teaches(x)

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication

hasConfPaper v hasPublication

PhDStudent v ≥2 hasConfPaper

Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication

q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)

⇓ unify the atoms

q3(x)← hasPublication(x , y)⇓ remove unbound variables

q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication

q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper

q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper

q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent

q7(x)← Student(x) ∧ E1teaches(x)

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering Algorithm

1 Compute the rewriting of the initial query, a UCQ.I Application of PIs to query atoms.

q(x)← hasPublication(x , y) ∧ Publication(y)

⇓ ≥1 hasPublication− v Publicationq′(x)← hasPublication(x , y) ∧ E1hasPublication

−(y)

I Unification of query atoms.

q(x)← hasPublication(x , y) ∧ E1hasPublication−(y)

⇓ unifyq′(x)← hasPublication(x , y)

2 Evaluate the obtained UCQ over the ABox viewed as a DB.

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite

• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′

I ≥k R v ≥k R ′ for each subrole R of R ′

• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R

• Unification for newly introduced predicates

I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)

I EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

• Horn inclusions increase the length of the query

I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite

• Number restrictions imply new inclusions:

extend the TBoxI ≥k R v ≥k ′ R, where k > k ′

I ≥k R v ≥k R ′ for each subrole R of R ′

• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R

• Unification for newly introduced predicates

I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)

I EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

• Horn inclusions increase the length of the query

I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite

• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′

I ≥k R v ≥k R ′ for each subrole R of R ′

• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R

• Unification for newly introduced predicates

I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)

I EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

• Horn inclusions increase the length of the query

I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite

• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′

I ≥k R v ≥k R ′ for each subrole R of R ′

• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R

• Unification for newly introduced predicates

I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)

I EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

• Horn inclusions increase the length of the query

I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite

• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′

I ≥k R v ≥k R ′ for each subrole R of R ′

• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R

• Unification for newly introduced predicates

I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)

I EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

• Horn inclusions increase the length of the query

I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite

• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′

I ≥k R v ≥k R ′ for each subrole R of R ′

• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R

• Unification for newly introduced predicates

I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)

I EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

• Horn inclusions increase the length of the query

I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite

• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′

I ≥k R v ≥k R ′ for each subrole R of R ′

• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R

• Unification for newly introduced predicates

I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)

I EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

• Horn inclusions increase the length of the query

I remove duplicated atoms

I remove Ek′R(z), if EkR(x) occurs in the query and k > k ′

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite

• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′

I ≥k R v ≥k R ′ for each subrole R of R ′

• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R

• Unification for newly introduced predicates

I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)

I EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

• Horn inclusions increase the length of the query

I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Complexity of the Algorithm

• Optimal data complexity: in AC0

• Combined complexity: in NP

Note that the size of the rewriting is exponentialalready w.r.t. the size of the TBox.

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Complexity of the Algorithm

• Optimal data complexity: in AC0

• Combined complexity: in NP

Note that the size of the rewriting is exponentialalready w.r.t. the size of the TBox.

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation

2 The DL DL-Lite(HN)horn

3 Knowledge Base Satisfiability

4 Query Answering

5 Conclusions

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Conclusion

• We reduced knowledge satisfiability and query answering in

DL-Lite(HN )horn to FOL evaluation.

I Practically implementable algorithms.I We can rely on relational database technology for managing the data

and query evaluation.

• The computational complexity of the algorithms is optimal w.r.t.data complexity:

I in AC0.

• Future work:I Implement the developed algorithms.I Study optimization techniques for the algorithm.I Extend the practical algorithm to positive existential queries.

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Conclusion

• We reduced knowledge satisfiability and query answering in

DL-Lite(HN )horn to FOL evaluation.

I Practically implementable algorithms.I We can rely on relational database technology for managing the data

and query evaluation.

• The computational complexity of the algorithms is optimal w.r.t.data complexity:

I in AC0.

• Future work:I Implement the developed algorithms.I Study optimization techniques for the algorithm.I Extend the practical algorithm to positive existential queries.

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Conclusion

• We reduced knowledge satisfiability and query answering in

DL-Lite(HN )horn to FOL evaluation.

I Practically implementable algorithms.I We can rely on relational database technology for managing the data

and query evaluation.

• The computational complexity of the algorithms is optimal w.r.t.data complexity:

I in AC0.

• Future work:I Implement the developed algorithms.I Study optimization techniques for the algorithm.I Extend the practical algorithm to positive existential queries.

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Motivation The DL DL-Lite(HN)horn

Knowledge Base Satisfiability Query Answering Conclusions

Thank youfor your attention!

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